Circular Motion

Rotating Turning about an internal axis Revolving Turning about an external axis

Linear speed How far you go in a certain amount of time Miles per hour, meters per second Rotational speed How many times you go around in a certain amount of time Revolutions per minute, rotations per hour

Which horse has a larger linear speed on a merry go round, one on the outside or one on the inside? Outside. Which horse has a greater rotational speed? Neither, all the horses complete the circle in the same amount of time.

How much faster will a horse at TWICE the distance from the center of the circle be moving TWICE the distance means TWICE the speed

The number of revolutions per second is called the frequency, f. Frequency is measured in Hertz, Hz. The time it takes to go all the way around once is called the period, T. Frequency is related to period by f = 1 / T and T = 1 / f

Example A boy twirls a toy airplane around and around at the end of a string. If it takes 2 seconds for the airplane to complete one loop, what is the frequency? f = 1 / T f = 1 / 2 f = 0.5 Hz

How do you find the velocity of an object moving in a circle if it is not directly provided? We know that Velocity = distance / time In circular motion, the distance traveled is all around the circle… the circumference. The circumference = 2  r The time it takes the object to go all the way around the circumference once is called the period, T. So… v = 2  r / T

Example A race car takes 1.5 minutes to go around one lap of a circular track. If the track has a radius of 400 m, how fast was the car traveling? v = 2  r / T v = 2  (400) / (1.5 x 60) v = 27.9 m/s

Uniform Circular Motion, UCM: moving in a circle with a constant speed. Question: Is there a constant velocity when an object moves in a circle with a constant speed? No, the direction changes, therefore the velocity changes. If the velocity changed, the object is actually ACCELERATING even while moving at the same speed.

Suppose an object was moving in a straight line with some velocity, v. According to Newton’s 1 st Law of Motion, “An object in motion continues that motion unless a net external force acts on it”. If you want the object to move in a circle, some force must push or pull it towards the center of the circle. A force that pushes or pulls an object towards the center of a circle is called a centripetal force Centripetal means “center seeking ”

What happens is the string breaks? Which way will the ball move? The ball will continue to move in a straight line path that is “tangent” to the circle.

According to Newton’s 2 nd Law, Fnet = ma, If there is a centripetal force, there must be a centripetal acceleration. a c = v 2 / r Where r is the radius of the circle and v is the velocity of the object.

Centripetal force Since F net = ma, the net centripetal force is given by

Lots of forces can help in pushing or pulling an object towards the center of a circle. Sometimes it takes more than one force to get an object to move in uniform circular motion. Centripetal force is NOT a new kind of force. If any force is making an object move in a circle, it becomes a centripetal force.

When can these forces be centripetal forces? Gravity? Moon revolving around the Earth Tension? Twirling a pail at the end of a string Friction? Cars rounding a curve. Air Resistance? Birds flying in a circle. Normal? Riders in a carnival ride

Example A boy twirls a ½ kg rock in a horizontal circle on the end of a 1.6 meter long string. If the velocity of the rock was 4 m/s, what is the Tension in the string? m = ½ kg r = 1.6 m v = 4 m/s The only centripetal force is Tension. T = m v 2 / r T = ½ 4 2 / 1.6 T = 5 N

Example How fast was the ½ kg rock moving if the Tension was 10 N and the string was 1.6 m long? m = ½ kg r = 1.6 m T = 10 N T = mv 2 / r Tr/m = v 2 10 x 1.6 /.5 = v 2 v = 5.7 m/s

Friction A 1500 kg race car goes around a circular track at 45 m/s. If the radius of the track is 100 m, how much friction is require to keep the car on the track? m = 1500 kg v = 45 m/s r = 100 m The centripetal force is friction. How much more friction is required if the velocity doubles? Triples?

Friction A 1500 kg race car goes around a circular track at 45 m/s. If the radius of the track is 100 m, how much friction is require to keep the car on the track? What is , the coefficient of friction? m = 1500 kg v = 45 m/s r = 100 m The centripetal force is friction. f = mv 2 /r f = 1500 x 45 2 / 100 f = N f =  N  = f / N N = mg = N  = N / N  = 2.02

Loop the Loop What is the minimum speed that a rider must be moving at in order to complete a loop the loop of radius 12 m? The weight is the only centripetal force when the rider is moving at the minimum required speed. mg = mv 2 /r g = v 2 /r v 2 = rg v 2 = 12 x 10 v = m/s

Problems 1.What is the minimum speed for a loop-the loop with radius 10 m? 8 m? 15 m? 2.A 1000 kg race car goes around a circular track at 35 m/s. If the radius of the track is 120 m, how much friction is require to keep the car on the track? What is , the coefficient of friction? 3. A 1200 kg race car goes around a circular track at 40 m/s. If the radius of the track is 110 m, how much friction is require to keep the car on the track? What is , the coefficient of friction?